42 2. FOUR IMPORTANT LINEAR PDE
THEOREM 16 (Uniqueness). There exists at most one solution u
C2(
¯
U ) of (46).
Proof. Assume ˜ u is another solution and set w := u ˜. u Then Δw = 0 in
U, and so an integration by parts shows
0 =
U
wΔw dx =
U
|Dw|2
dx.
Thus Dw 0 in U, and, since w = 0 on ∂U, we deduce w = u ˜ u 0 in
U.
b. Dirichlet’s principle. Next let us demonstrate that a solution of the
boundary-value problem (46) for Poisson’s equation can be characterized as
the minimizer of an appropriate functional. For this, we define the energy
functional
I[w] :=
U
1
2
|Dw|2
wf dx,
w belonging to the admissible set
A := {w
C2(
¯
U ) | w = g on ∂U}.
THEOREM 17 (Dirichlet’s principle). Assume u
C2(
¯
U ) solves (46).
Then
(47) I[u] = min
w∈A
I[w].
Conversely, if u A satisfies (47), then u solves the boundary-value problem
(46).
In other words if u A, the PDE −Δu = f is equivalent to the statement
that u minimizes the energy I[ · ].
Proof. 1. Choose w A. Then (46) implies
0 =
U
(−Δu f)(u w) dx.
An integration by parts yields
0 =
U
Du · D(u w) f(u w) dx,
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